COMCOT User Manual v1.7

7/25/2019 COMCOT User Manual v1.7

1/65

USER MANUAL FOR COMCOT VERSION 1.7

(FIRST DRAFT)

by

Xiaoming Wang

Febury 2009


2/65

ACKNOWLEDGEMENTS

COMCOT version 1.7 was developed by Xiaoming Wang at Institute of Geological &

Nuclear Science, New Zealand. Part of work and all its earlier versions were accom-

plished by Xiaoming Wang, Y.-S. Cho, S.-B. Woo and other members in the Wave Group

led by Professor P. L.-F. Liu at Cornell University, USA.

This document serves as an improvement and appendix to Computer programs for

tsunami propagation and inundation, edited by Liu, P. L.-F., Woo, S.-B. and Cho, Y.-S.

(1998), Cornell University.

The author takes no responsibilties on any type of actual or potential damages or

losses caused by using the entire or any part of this code.

2


3/65

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Introduction 1

1.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Governing Equations and Numerical Methods 5

2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Nested Grid Configuration . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Moving Boundary Scheme . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Dispersion Improvement . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Tsunami Generation 21

3.1 Sea Floor Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Instantaneous Deformation - Elastic Fault Model . . . . . . . . 22

3.1.2 Transient Seafloor Motion . . . . . . . . . . . . . . . . . . . . 24

3.2 Water Surface Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Parameter Configuration, Input and Output Data 28

4.1 Control Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.1 General Information . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.2 Parameters for Fault Model . . . . . . . . . . . . . . . . . . . . 31

4.1.3 Parameters for Wave Maker . . . . . . . . . . . . . . . . . . . 33

4.1.4 Parameters for Landslides/Transient Floor Motion . . . . . . . 34

4.1.5 Parameters for All Grid Levels . . . . . . . . . . . . . . . . . . 35

4.2 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Bathymetric/Topographical Data . . . . . . . . . . . . . . . . . 40

4.2.2 Seafloor Deformation Data . . . . . . . . . . . . . . . . . . . . 41

4.2.3 Transient Sea Floor Motion Data . . . . . . . . . . . . . . . . . 43

4.2.4 Bottom Friction Coefficients . . . . . . . . . . . . . . . . . . . 45

4.2.5 Time History Input for Wave Maker . . . . . . . . . . . . . . . 45

4.2.6 Numerical Tidal Gauge Locations . . . . . . . . . . . . . . . . 454.3 Output Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.1 Time Sequence Data . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.2 Water Surface Elevation/Volume Fluxes . . . . . . . . . . . . . 46

4.3.3 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.4 Seafloor Deformation . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.5 Maximum Water Surface Elevation/Depression . . . . . . . . . 49

3


4/65

4.3.6 Time History Records . . . . . . . . . . . . . . . . . . . . . . 50

4.3.7 Hot Start/Automatic Data Backup . . . . . . . . . . . . . . . . 51

A Flow Chart of COMCOT 53

B Oblique Stereographic Projection 54

C MatLab Scripts for Data Processing 57

4


5/65

LIST OF TABLES

3.1 Parameters for Elastic Fault Plane Model . . . . . . . . . . . . . . . . 22

4.1 Fortran scripts to write Water surface/Volume flux data into a data file . 47

4.2 Fortran scripts to write Initial Condition into a data file . . . . . . . . . 484.3 Fortran scripts to write Seafloor Deformation into a data file . . . . . . 49

4.4 Fortran scripts to write Max/Min Water Surface Fluctuations into a data

file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Fortran scripts to write snapshots for hot start . . . . . . . . . . . . . . 51

5


6/65

LIST OF FIGURES

2.1 Sketch of Staggered Grid Setup in COMCOT. . . . . . . . . . . . . . . 8

2.2 Sketch of Nested Grid Setup . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Detailed view of Nested Grid Setup. Left panel: grid nesting at lower-

left corner of sub-level grid region; Right panel: grid nesting at upper-

right corner of sub-level grid region. . . . . . . . . . . . . . . . . . . . 13

2.4 Detailed Time Marching Scheme of Nested Grid Setup. . . . . . . . . 16

2.5 A sketch of moving boundary scheme. . . . . . . . . . . . . . . . . . 18

3.1 Sketch of a Fault Plane and Fault parameter definitions. . . . . . . . . 26

3.2 Sketch of Transient Sea Floor Motion). . . . . . . . . . . . . . . . . . 27

4.1 General Parameter Section inCOMCOT.CT L . . . . . . . . . . . . . 29

4.2 Fault Parameter Section inCOMCOT.CT L . . . . . . . . . . . . . . . 31

4.3 Wave Maker Parameter Section inCOMCOT.CT L . . . . . . . . . . . 33

4.4 Parameters for 1st-level grid inCOMCOT.CT L . . . . . . . . . . . . . 364.5 Parameters for sub-level grids inCOMCOT.CT L . . . . . . . . . . . . 39

4.6 Example of Bathymetry Data File (XYZ format) . . . . . . . . . . . . 40

4.7 Example of Seafloor Deformation Data File (XYZ Format) . . . . . . . 42

4.8 Example of Landslide Data File (XYT Format) . . . . . . . . . . . . . 44

4.9 Example of Tidal Gauge Location File ts location.dat . . . . . . . . . 46

A.1 Flow Chart of COMCOT. . . . . . . . . . . . . . . . . . . . . . . . . 53

6


7/65

CHAPTER 1

INTRODUCTION

Tsunami is a typical long wave in the ocean, generated by seafloor or water surface dis-

turbances over a sufficiently large area. It has been observed and recorded since ancient

times, especially in Japan and the Mediterranean areas. The earliest recorded tsunami

occurred in 2, 000 B.C. offthe coast of Syria (Lander and Lockridge, 1989). In recent

years, the most devastating tsunami was triggered by the 2004 Sumatra earthquake o ff

the coast of Indonesia and caused tremendous property loss and over 225, 000 casual-

ties in the surrounding countries of Indian Ocean, especially in Indonesia, Sri Lanka,

Thailand and India.

For the tsunami hazard mitigation, it is very important to construct inundation maps

along those coastlines vulnerable to tsunami flooding. These maps should be developed

based on the historical tsunami events and hypothetical scenarios. To produce realistic

and reliable inundation estimates, it is essential to use a numerical model that calculates

accurately the tsunami propagation from a source region to the coastal areas of concern

and the subsequent tsunami run-up and inundation.

In many events, the wavelength of a tsunami is usually long compared to the wa-

ter depth, the dispersion effect, measured by = h0/l0, can be practically neglected if

h0/l0 is smaller than 1/20. In this scenario, shallow water equations are adequate for

studying tsunami evolutions. Numerical models based on shallow water equations are

generally very efficient in simulating transoceanic tsunamis due to the use of explicit

numerical schemes and no need to solve higher-order derivatives associated with non-

linearity and frequency dispersion. COMCOT is one of the models based on Shallow

Water Equations.

1


8/65

COMCOT (Cornell Multi-grid Coupled Tsunami model) adopts explicit staggered

leap-frog finite difference schemes to solve Shallow Water Equations in both Spheri-

cal and Cartesian Coordinates. A nested grid system, dynamically coupled up to 12

levels (which will be also referred to as layers) with different grid resolution, can be

implemented in the model to fulfill the need for tsunami simulations in different scales.

Nested grid system means in a region of one grid size, there is one or more regions with

smaller grid sizes situated in, which eventually, forms a hierarchy of grids, or grid levels.

The region with largest grid size is called 1st-level grid and all the grid regions directly

nested in the 1st level grid, are called 2nd-level grids, so on and so forth. In one grid

region, up to 12 sub-level grid regions can be defined. It should be noted that in one grid

region, a uniform grid size (x = y) is adopted in COMCOT. Spherical or Cartesian

coordinate system, as well as either linear or nonlinear version of governing equations

can be chosen for each region. The water surface displacement is assumed the same as

the deformation of the sea floor as long as the uplift motion is much faster than the wave

propagation; otherwise, submarine landslide model should be used to include transient

effect. For a given earthquake, the displacement of seafloor is determined from a linear

elastic dislocation theory (Mansinha and Smylie, 1971; Okada, 1985).

For theoretical background and detailed derivations, please refer to Liu et al. (1998),

Computer programs for tsunami propagation and inundation, Cornell University.

1.1 Features

COMCOT is capable of efficiently studying the entire life-span of a tsunami, including

its generation, propagation, runup and inundation. Multiple tsunami-generating mech-

anisms can be implemented in COMCOT, including instal or transient faulting, land-

2


9/65

slides, water surface disturbances or wave maker. Both linear and nonlinear Shallow

Water Equations are available in spherical or Cartesian coordinate system for numer-

ical studies at different scales. Nested grid configuration balances the efficiency and

accuracy in which larger grid size can be used in the open sea for studying tsunami

propagation and finer grids can be adopted for coastal regions of interest. Some features

are briefly listed below:

Explicit Leap-Frog finite difference scheme to solve linear and nonlinear shallow

water equations in spherical and cartesian coordinates;

Nested grid configuration

Multiple tsunami-generating mechanisms, including instant faulting ( Mansinha

and Smylie (1971) and Okada (1985)), transient floor motion (e.g., landslides or tran-

sient faulting), water surface disturbance and wave maker;

Implementation of constant or variable bottom roughness;

Dispersion-improved numerical scheme;

Flexible input/output options

1.2 Brief History

COMCOT originated from the creation of Yongsik Cho and S.N. Seo (version 1.0)

based on the theoretical and numerical work of Shuto (1991);? and Imamura et al.

(1988). Moving boundary scheme was introduced and validated against experimental

studies (Liu et al., 1995). Dr. Seung-Buhm Woo introduced a user interface (i.e., pa-

rameter file) and develop a general grid matching algorithm (1999) and finally formed

3


10/65

version 1.4. Many achievements were made with COMCOT during this period, such as

the successful simulations of 1960 Chile Tsunami (Liu et al., 1994) and 1986 Taiwan

Hua-lien tsunami, involving the modelling of tsunami generation, propagation, runup

and inundation. The coming of Fortran 90 gave a new power to the programming of

COMCOT. With the helps of many others, especially Tom Logan (ARSC), Steven Lantz

(Cornell University) and Philip L.-F. Liu (Cornell University), further improvements and

modifications were introduced by Xiaoming Wang (2003-), which yielded version 1.5

and 1.6. And subroutines dealing with nonlinear equations were also optimized to get

a better efficiency by Tom Logan (ARSC, 2003). One of the significant progresses is

the migration from Fortran 77 to Fortran 90, allowing dynamic allocation of arrays.

The structure of COMCOT was re-organized and a new user interface and parameter

modules was introduced, making the code much neat and more expandable. Most of

subroutines were rewritten during this time and more tsunami-generating mechanism

were implemented. A new grid-nesting algorithm was developed to allow more effi-

cient and flexible grid setup. Numerical simulations performed on this version include

2002 Hua-lien tsunami, 2003 Algerian tsunami, 2004 and 2005 Indian Ocean tsunamis

and 2006 Java tsunami. For the 2004 Giant Tsunami, both runup and inundation were

extensively studied in several regions.

In version 1.7, the solver kernel is re-packaged with a more efficient interface. More

data formats are supported. Moreover, the grid-matching becomes much easier. Once

the region of sub-level grids is defined (by coordinates of lower-left and upper-right cor-

ners), the grid-matching is done automatically by COMCOT. In addition, a dispersion-

improved algorithm is also added into the solver so that COMCOT will be able to sim-

ulate linear/weakly nonlinear dispersive waves, but still keep a high computational effi-

ciency.

4


11/65

CHAPTER 2

GOVERNING EQUATIONS AND NUMERICAL METHODS

2.1 Governing Equations

Linear and nonlinear shallow water equations in both Spherical and Cartesian Coordi-

nates are implemented in COMCOT. For tsunamis in deep ocean, tsunami amplitude

is much smaller than the water depth and linear shallow water equations in Spherical

Coordinates can be applied.

t+

1

R cos

P

+

(cos Q)

=

h

t(2.1)

P

t+

gh

R cos

f Q= 0 (2.2)

Q

t+

gh

R

+ f P = 0 (2.3)

where is the water surface elevation; (P, Q) denote the volume fluxes in X (West-

East) direction and Y(South-North) direction, respectively; (, ) denote the latitude

and longitude of the Earth;Ris the radius of the Earth;gis the gravitational acceleration

and h is the water depth. And the termht

reflects the effect of transient seafloor motion,

can be used to model landslide-generated tsunamis. f represents the Coriolis force

coefficient due to the rotation of the Earth and

f = sin (2.4)

and is the rotation rate of the Earth.

When a simulation involving a relatively small region in which the Earth rotation

effect is not prominent, shallow water equations in Cartesian Coordinates are preferred.

The follow linear shallow water equations in Cartesian Coordinates are also imple-

5


12/65

mented in COMCOT.

t+

P

x+Q

y

=

h

t(2.5)

Pt

+ ghx

f Q = 0 (2.6)

Q

t+ gh

y+ f P = 0 (2.7)

where (P, Q) denote the volume fluxes in X(West-East) direction and Y (South-North)

direction, respectively, and both are products of velocity and water depth, i.e., P = hu

and Q = hv.

Since as the tsunami propagates over a continental shelf and approaches a coastal

area, linear shallow water equations are no longer valid. The wave length of the incident

tsunami becomes shorter and the amplitude becomes larger as the leading wave of a

tsunami propagates into shallow water. Therefore, the nonlinear convective inertia force

and bottom friction terms become increasingly important, while the significance of the

Coriolis force and the frequency dispersion terms diminishes. The nonlinear shallow

water equations including bottom friction effects are adequate to describe the flow mo-

tion in the coastal zone (Kajiura and Shuto, 1990; Liu et al., 1994). Furthermore, along

the shoreline, where the water depth becomes zero, a special treatment is required to

properly track shoreline movements.

In COMCOT, the following nonlinear shallow water equations are implemented in

Spherical Coordinates as

t

+ 1R cos

P

+

(cos Q)

= ht

(2.8)

P

t+

1

R cos

P2

H

+

1

R

PQ

H

+

gH

R cos

f Q + Fx =0 (2.9)

Q

t+

1

R cos

PQ

H

+

1

R

Q2

H

+

gH

R

+ f P + Fy =0 (2.10)

6


13/65

and in Cartesian Coordinates as

t+

P

x+Q

y

=

h

t(2.11)

Pt

+ x

P

2

H

+ y

PQH

+ gH

x+ Fx =0 (2.12)

Q

t+

x

PQ

H

+

y

Q2

H

+ gH

y+ Fy =0 (2.13)

in which H is the total water depth and H = + h; Fx and Fy represents the bottom

friction in X and Y directions, respectively. And these two terms are evaluated via

Mannings formula

Fx = gn2

H7/3 P(P2 + Q2)1/2 (2.14)

Fx = gn2

H7/3Q(P2 + Q2)1/2 (2.15)

wheren is the Mannings roughness coefficient.

2.2 Numerical Method

Explicit leap-frog finite difference method is adopted to solve both linear and nonlinear

shallow water equations (Cho, 1995). The evaluation of water surface elevation,, and

volume fluxes, P and Q, are staggered in time and space. Figure 2.1 shows a grid

system in which the water surface displacement is evaluated at the center of a grid cell

and volume fluxes P and Q (product of velocity and water depth) are evaluated at edge

centers (i.e., (i + 1/2, j) and (i, j + 1/2)) of the grid cell.

The numerical scheme has a truncation error ofO(x2,y2,t2). The explicit leap-

frog finite difference scheme for linear shallow water equations can be expressed as

n+1/2

i,j

n1/2

i,j

t+

1

R cos

i,j

Pni+1/2,j

Pni1/2,j

(2.16)

7


14/65

Figure 2.1: Sketch of Staggered Grid Setup in COMCOT.

+

1

R cos

i,j

(cos i,j+1/2)Qni,j+1/2

(cos i,j1/2)Qni,j1/2

=

hn+1/2

i,j h

n1/2

i,j

t(2.17)

Pn+1i+1/2,j P

ni+1/2,j

t+

ghR cos

i+1/2,j

n+1/2

i+1,j n+1/2

i,j

f Qni+1/2,j =0 (2.18)

Qn+1i,j+1/2

Qni,j+1/2

t+

gh

R

i,j+1/2

n+1/2

i,j+1

n+1/2

i,j

+ f Qni,j+1/2 =0 (2.19)

in Spherical Coordinates for a large scale simulation and

n+1/2

i,j

n1/2

i,j

t+

Pni+1/2,j

Pni1/2,j

x+

Qni,j+1/2

Qni,j1/2

y=

hn+1/2

i,j h

n1/2

i,j

t(2.20)

Pn+1i+1/2,j

Pni+1/2,j

t + ghi+1/2,j

n+1/2

i+1,j

n+1/2

i,j

x =0 (2.21)

Qn+1i,j+1/2

Qni,j+1/2

t+ ghi,j+1/2

n+1/2

i,j+1

n+1/2i,j

y=0 (2.22)

in Cartesian Coordinates for a small scale simulation.

From the continuity equation, the proposed leap-frog scheme calculates the free sur-

8


15/65

face elevation at the (i, j) thgrid point on the (n + 1/2) thtime step. These computa-

tions are fully explicit and require information on the volume flux components and the

free surface displacement from the previous time step. The volume flux components are

not evaluated at the same location as that for the free surface displacement. Figure ??

shows a grid system in which the free surface displacement, , is calculated at the center

of a grid cell (i, j) and the volume flux components, P and Q, are obtained at the centers

of four edges of the grid cell, i.e., Pi1/2,j, Pi+1/2,j, Q i,j1/2 and Qi,j+1/2. The momentum

equations, ?? and ??, are then used to calculate the volume flux components, Pi+1/2,jand

Qi,j+1/2. Note that the calculations for the free surface displacement and the volume flux

components are also staggered in time.

The nonlinear shallow water equations are discretized by using the same leap-frog

finite difference scheme as the linear shallow water equations. The nonlinear convection

terms are discretized with an upwind scheme. In general, the upwind scheme is condi-

tionally stable and introduces some numerical dissipation. But if the velocity gradient

in the fluid field is not too large and if the stability condition, which is

ght/x < 1

is satisfied, upwind formulation is preferred for solving advective terms since, at each

time step, only a small computational effort is required. The free surface elevation is

evaluated at time levels t = (n 1/2)tand t = (n+ 1/2)thowever the volume flux

components are calculated at time levels t=ntand t= (n + 1)t.

The nonlinear convective terms in the momentum equations in Cartesian coordinate

system are discretized by using an upwind scheme and given as

x

P2

H

=

1

x

11 (Pni+3/2,j)2

Hni+3/2,j

+ 12(Pni+1/2,j)2

Hni+1/2,j

+ 13(Pni1/2,j)2

Hni1/2,j

(2.23)

y

PQ

H

=

1

y

21(PQ)n

i+1/2,j+1

Hni+1/2,j+1

+ 22(PQ)n

i+1/2,j

Hni+1/2,j

+ 23(PQ)n

i+1/2,j1

Hni+1/2,j1

(2.24)

x

PQ

H

=

1

x

31(PQ)n

i+1,j+1/2

Hni+1,j+1/2

+ 32(PQ)n

i,j+1/2

Hni,j+1/2

+ 33(PQ)n

i1,j+1/2

Hni1,j+1/2

(2.25)

9


16/65

y

Q2

H

=

1

y

41(Qn

i,j+3/2)2

Hni,j+3/2

+ 42(Qn

i,j+1/2)2

Hni,j+1/2

+ 43(Qn

i,j1/2)2

Hni,j1/2

(2.26)in which the coefficient,, are determined from

11 =0, 12 =1, 13 =1, i f P

ni+1/2,j

0

11 =1, 12 = 1, 13 =0, i f Pni+1/2,j


17/65

Pn+1i+1/2,j = 1

1 + xt

(1 xt)P

ni+1/2,j rxgH

n+1/2

i+1/2,j(

n+1/2

i+1,j

n+1/2

i,j )

rx

1 + xt

11

(Pni+3/2,j

)2

Hni+3/2,j

+ 12(Pn

i+1/2,j)2

Hni+1/2,j

+ 13(Pn

i1/2,j)2

Hni1/2,j

rx

1 + xt

21 (PQ)ni+1/2,j+1

Hni+1/2,j+1

+ 22(PQ)ni+1/2,j

Hni+1/2,j

+ 23(PQ)ni+1/2,j1

Hni+1/2,j1

(2.36)Qn+1i,j+1/2 =

1

1 + yt

(1 yt)Q

ni,j+1/2 rygH

n+1/2

i,j+1/2(

n+1/2

i,j+1

n+1/2

i,j )

ry

1 + yt

31(PQ)n

i+1,j+1/2

Hni+1,j+1/2

+ 32(PQ)n

i,j+1/2

Hni,j+1/2

+ 33(PQ)n

i1,j+1/2

Hni1,j+1/2

ry

1 + yt

41

(Qni,j+3/2

)2

Hni,j+3/2

+ 42(Qn

i,j+1/2)2

Hni,j+1/2

+ 43(Qn

i,j1/2)2

Hni,j1/2

(2.37)

in which rx = t/x and ry = t/y. Following approximations have been used to

derive the above finite difference equations.

Hn+1/2

i+1/2,j=

1

2(H

n+1/2

i,j + H

n+1/2

i+1,j ) (2.38)

Hn+1/2

i,j+1/2=

1

2(H

n+1/2i,j

+ Hn+1/2

i,j+1 ) (2.39)

Hni+1/2,j = 1

4(H

n1/2

i,j + H

n+1/2

i,j + H

n1/2

i+1,j + H

n+1/2

i+1,j ) (2.40)

Hn

i,j+1/2 =

1

4 (Hn1/2

i,j + H

n+1/2

i,j + H

n1/2

i,j+1 + H

n+1/2

i,j+1 ) (2.41)

Pni,j+1/2 = 1

4(Pni1/2,j+ H

ni1/2,j+1+ P

ni+1/2,j + P

ni+1/2,j+1) (2.42)

Qni+1/2,j = 1

4(Qni,j1/2+ Q

ni+1,j1/2+ Q

ni,j+1/2+ Q

ni+1,j+1/2) (2.43)

In spherical coordinates, the nonlinear shallow water equations are discretized as

n+1/2

i,j =

n1/2

i,j

1

R cos i,j

t

(Pni+1/2,j P

ni1/2,j)

1

R

t

(Qni,j+1/2 Q

ni,j1/2) (2.44)

Pn+1i+1/2,j = fx

(1 xt)Pni+1/2,j

gH

R cos

n+1/2i+1/2,j

t

(

n+1/2

i+1,j

n+1/2i,j

)

fx

R cos i,j

t

11(Pn

i+3/2,j)2

Hni+3/2,j

+ 12(Pn

i+1/2,j)2

Hni+1/2,j

+ 13(Pn

i1/2,j)2

Hni1/2,j

11


18/65

fx

R

t

21(PQ)n

i+1/2,j+1

Hni+1/2,j+1

+ 22(PQ)n

i+1/2,j

Hni+1/2,j

+ 23(PQ)n

i+1/2,j1

Hni+1/2,j1

(2.45)

Qn+1i,j+1/2 = fy

(1 yt)Q

ni,j+1/2

gHn+1/2

i,j+1/2

R

t

(

n+1/2

i,j+1

n+1/2

i,j )

fy

R cos i,j+1/2

t

31(PQ)n

i+1,j+1/2

Hni+1,j+1/2

+ 32(PQ)n

i,j+1/2

Hni,j+1/2

+ 33(PQ)n

i1,j+1/2

Hni1,j+1/2

fy

R

t

41(Qn

i,j+3/2)2

Hni,j+3/2

+ 42(Qn

i,j+1/2)2

Hni,j+1/2

+ 43(Qn

i,j1/2)2

Hni,j1/2

(2.46)

in which fx = 1

1+xtand fy =

11+yt

. Other variables are the same as defined before.

2.3 Nested Grid Configuration

As we know during the shoaling of waves, their length will become much smaller than

in deep ocean. Finer grids will be necessary to resolve wave profiles in shallow water

regions. Therefore, when the water depth varies within the computational domain, it

might be desirable that different grid size and time step size be employed in different

sub-regions. The nested grid configuration allows to obtain detailed information in the

coastal region. Finer grids should be used only in specific regions of interests.

In COMCOT, either the linear or nonlinear shallow water equations with either

spherical or cartesian coordinate system can be assigned to a specific sub-level region.

These sub-level grid regions are dynamically connected. The any ratio of grid size be-

tween two directly nested grid layers can be used, however it should be an integer.

Here we briefly describe the technique for exchanging information between two

nested grid layers of different grid sizes. As shown in Figure 2.2 and 2.3, a smaller grid

system is nested in a larger grid system with the ratio of 1 : 3. The arrows represent

the volume fluxes, P and Q, across each grid cell, while squares and dots indicate the

12


19/65

Figure 2.2: Sketch of Nested Grid Setup

Figure 2.3: Detailed view of Nested Grid Setup. Left panel: grid nesting at lower-

left corner of sub-level grid region; Right panel: grid nesting at upper-

right corner of sub-level grid region.

locations where the free surface displacement is evaluated.

At a certain time level, volume fluxes in both large and small grid systems are de-

termined from the momentum equations, with the exception of volume fluxes for the

smaller grid system along the boundaries between two grid regions. These data are de-

13


20/65

termined by interpolating the neighboring volume fluxes from the large grid system. The

free surface displacement at the next time level for the small grid system can be calcu-

lated from the continuity equation. Usually the time step size for the smaller grid system

is also smaller than that used in the larger grid system to sastify the CFL (Courant-

Friedrichs-Lewy) condition. Therefore, the volume fluxes along the boundary of the

small grid system at the next time level must be obtained by interpolating the neighbor-

ing volume fluxes obtained from the large grid system over a larger time interval. After

the free surface displacements in the small grid system are calculated up tot he next time

level of the large grid system, the free surface displacements in the large grid system are

updated by solving the continuity equation.

Let us describe these procedure step by step. Suppose all flux values in the inner

and the outer region are known at time level t=ntand we need to solve the inner and

the outer region values at the next time step t = (n +1)t. Since the outer grid region

and the inner grid region adopt different grid sizes, the time step sizes for each region

are different due to the requirement of stability. In COMCOT, the ratio of time step size

between the outer grid region and inner grid region varies, depending on the maximum

water depth in each region. And for simplicity, also assume that the time step of the

inner region is one half of that of the outer region (see Figure 2.4).

*Get the free surface elevation at t = (n + 1/2)tin the outer region by solving

continuity equation.

*To solve the continuity equation in the inner region, however, we need to have the

flux information along the connected boundary at t=nt. So the flux values in the outer

grids at the connected boundary are linearly interpolated and then those interpolated

values are assigned to the fluxes in the inner at the boundary.

14


21/65

*Get the free surface elevation att=(n + 1/4)tin the inner grid region by solving


*Get the flux values att=(n + 1/2)tin the inner grid region by solving momentum

equation.

*To get the free surface elevation at t = (n + 3/4)tin the inner grid region, we

should have the flux information along the connected boundary at t = (n + 1/2)t. To

get these information we can do the in the following way. First, since we already know

the free surface elevation att= (n+1/2)tand flux att= ntin the outer region, we can

get the flux in the outer region along the connected boundary at t= (n + 1t) by solving

linear momentum equation locally. Second, these flux values at t= (n + 1)tare linearly

interpolated along the connected boundary. And to get the value att= (n + 1/2)t, outer

flux values att= ntandt=(n +1)tare also linearly interpolated. Those spatially and

timely interpolated flux values are assigned to the flux in the inner grid at the boundary.

*Get the free surface elevation att=(n + 3/4)tin the inner grid region by solving


*To transfer the information from the inner grid region to the outer region, the free

surface elevation in the inner grid region is spatially averaged over the grid size of the

outer region. These averaged elevation values at t=(n + 3/4)tare then time averaged

with those at t = (n+ 1/4)tin the inner region. These spatially and timely averaged

elevation values in the inner grid region update the elevation values att= (n + 1/2)tin

the outer region.

*Get the flux values at t = (n + 1)tin the inner region by solving momentum

equation;

15


22/65

*Get the flux values at t = (n + 1)tin the outer region by solving momentum

equation.

The above scheme of time and space evoluation for nested grids is illustrated in

Figure 2.4 for 1-D problem. This nested grid scheme has been validated and applied to

Figure 2.4: Detailed Time Marching Scheme of Nested Grid Setup.

several numerical, experimental and practical studies (Liu et al., 1994, 1995; Cho, 1995;

Liu et al., 1998; Wang and Liu, 2005, 2007).

2.4 Moving Boundary Scheme

In carrying out numerical computations, the computational domain is divided into finite

difference grids. Initially, the free surface displacement is zero everywhere, as are the

volume fluxes. When the grid point is on dry land, the water depth,h, takes a negative

value and gives the elevation of the land measured from the Mean Water Level (MWL).

Figure 2.5 shows a schematic sketch of the moving boundary treatment used in the

16


23/65

study (Cho, 1995; Liu et al., 1995). The MWL represents the mean water level and Hf

denotes the flooding depth in Figure 2.5. In a land (dry) cell the total depth, H=h + ,

has a negative value. On the other hand, the wet cell has a positive H value. The

continuity equation in conjunction with boundary conditions along offshore boundaries

is used to find free surface displacements at the next time step in the entire computational

domain, including the dry (land) cells. The free surface displacement at a dry land grid

remains zero because the volume fluxes are zero at the neighboring grid points (See

figure 2.5). At a shoreline grid, the total depth,H, is updated. A numerical algorithm

is needed to determine if the total water depth is high enough to flood the neighboring

dry (land) cells and hence to move the shoreline. The momentum equations are used to

update the volume fluxes in the wet cells only.

To illustrate the moving boundary algorithm, the one-dimensional case is used as an

example. As shown in Figure 2.5, the real bathymetry has been replaced by a staircase

representation. The total depth, H, is calculated and recorded at grid points i 1, i and

i + 1, while the volume flux is computed at grid points i 1/2,i + 1/2 and i + 3/2. As

shown in Figure 2.5), thei th cell is a wet cell in which the total depth is positive and

the (i + 1) thcell is dry (land) cell in which the total depth is negative and the volume

fluxes are zero. The shoreline is somewhere between the i th cell and (i + 1) th

grid points. Then, the volume flux at the (i 1/2) thgrid point is assigned to be zero.

Therefore, the shoreline does not move to the onshore direction. When the water surface

is rising as shown in Figure 2.5, however, the volume flux at the ( i + 1/2) thgrid point

is no longer zero. The shoreline may move one grid point to the onshore direction. After

the total depth has been updated from the continuity equation, the following algorithm

is used to determine whether or not the shoreline should be moved. IfHi > 0, possible

cases can be summarized as

17


24/65

Figure 2.5: A sketch of moving boundary scheme.

* IfHi+1 0 andhi+1+ i 0, then the shoreline remains between grid pointsi and

i + 1 and the volume flux Pi+1/2remains zero;

* IfHi+1 0 andhi+1 + i >0, then the shoreline moves to between grid points i + 1

andi + 2 and the volume flux Pi+1/2may have a nonzero value, whilePi+3/2 is assigned

to be zero. The flooding depth is Hf =hi+1+ i;

* IfHi+1 > 0, then the shoreline moves to between grid points i + 1 and i + 2. The

volume fluxPi+1/2may have a nonzero value while Pi+3/2has a zero value. The flooding

depth isHf =max(hi+1+ i, hi+1+ i+1).

In the above cases, the time-step index has been omitted for simplicity. The algo-

18


25/65

rithm is developed for a two-dimensional problem and the corresponding y direction

algorithm has the same procedure as that for the x direction.

To save computing time, the regions that represent permanent dry (land) can be ex-

cluded from the computation by installing a depth criterion. In COMCOT, grid cells

with land elevation greater than 50.0 meters, will not be included in the computation.

Moreover, whenHis very small, the associated bottom friction term become very large

and, accordingly, a lower bound of the water depth is used to avoid the difficulty. The fi-

nite difference approximation for the continuity equation correctly accounts for positive

and zero values of the total depth on each side of a computational grid. The treatment

of flooding and ebbing grid cells guarantees mass conservation while accounting for the

flooding and ebbing of land. The occurrence of a zero value for the total depth Hon one

side of a cell implies zero mass flux until Hbecomes positive. A grid cell is considered

a dry cell only if the total water depths at all sides are zero or negative. This moving

boundary scheme has been validated and applied to several experimental and practical

studies (Liu et al., 1994, 1995; Cho, 1995; Wang and Liu, 2007).

2.5 Dispersion Improvement

For most tsunami events, length scales of tsunamis are much larger than water depth over

which they propagate. Therefore, physical frequency dispersion effect can be neglected

and Shallow Water Equations are adequate enough to describe tsunami waves without

considering dispersion. However, the wavelength of a tsunami may not be large enough

compared to water depth so that the dispersion effect can be neglected. Moveover, the

dispersion effect is also accumulated over distance. Thus, in some conditions, the dis-

persion effect should be considered in a simulation. Physically, Boussinesq Equation

19


26/65

models or 3-D Navier-Stokes equation models are required to account for the dispersion

effects. But considering the scale of a tsunami simulation and computational cost of

these models, Boussinesq Equation models or 3-D Navier-Stokes equation models are

not a preferable choice. As we know, numerical dispersion is an inherent phenomena

in discretizing shallow water equation with explicit leap-frog finite difference scheme.

Imamura et al. (1988) proposed a method to mimic physical dispersion by using this

type of numerical dispersion for linear shallow water equation models over constant

water depth. By choosing grid size according to the following relationship,

x = 4h2 + gh(t)2 (2.47)

in which h is water depth; g is gravitational acceleration and t is the size of time

step, then physical dispersion can be recovered. Cho (1995) improved the method to

correct the accuracy in diagonal directions. Yoon (2002) further improve the scheme to

describe the dispersion effect over a slowly varying bathymetry. Later on, Wang (2008)

extended the idea to weakly nonlinear weakly dispersive waves over a slowly varying

bathymetry so that the numerical dispersion in the shallow water equation model can be

used to recover the dispersion relationship of traditional Boussinesq equations.

Please refer to Wang (2008) for more details.

20


27/65

CHAPTER 3

TSUNAMI GENERATION

Tsunamis are generated by large-area disturbances on seafloor, such as submarine earth-

quakes, landslides or volcanic eruptions, or on water surface such as meteorite im-

pacts. Multiple tsunami generation mechanisms can be implemented in COMCOT, in-

cluding instantaneous seafloor rupture computed via fault models, e.g., Okada (1985)

and Mansinha and Smylie (1971), transient floor motions (e.g., transient ruptures, land-

slides), customized profiles of water surface displacements and wave makers.

3.1 Sea Floor Disturbances

Depending on the duration of the seafloor motion relative to the wave period of tsunamis,

the seafloor disturbances can be classified into two categories. If the duration of a floor

motion is much shorter the tsunami wave period being generated, the motion can be as-

sumed instantaneous (i.e., a sudden uplift of seafloor), then the displacement of seafloor

can be estimated via fault models (Mansinha and Smylie, 1971; Okada, 1985) and the

water surface is assumed to exactly follow the uplift the seafloor since water over this

large rupture area can not be drained out within such short duration. However, if the

duration of the floor rupture motion is comparable to tsunami wave period, the rupture

should be considered as a transient process and this transient process should be modelled

in tsunami simulations.

21


28/65

3.1.1 Instantaneous Deformation - Elastic Fault Model

For instantaneous seafloor rupture, the seafloor displacement caused by an earthquake

event is computed via elastic finite fault plane theory proposed originally by Mansinha

and Smylie (1971) and then improved by Okada (1985). Both models are available in

COMCOT. The theory assumes a rectangular fault plane being buried in semi-infinite

elastic half plane. This plane is an idealized representation of the interface between two

colliding tectonic plates where violent relative motion (i.e., dislocation) occurs during an

earthquake event (see Figure 3.1). The dislocation (or slip motion) occurring on the fault

plane will then deform the surface of the semi-infinite medium, which is considered as

the seafloor displacement during the earthquake event. To compute the deformation, the

following fault parameters are necessary. The definition of these parameters can be seen

Table 3.1: Parameters for Elastic Fault Plane Model

Parameters Unit

Epicenter (Lat, Lon) Degrees

Focal depth Meters

Length of Fault Plane Meters

Width of Fault Plane Meters

Dislocation (slip ) Meters

Strike direction () Degrees

Dip angle () Degrees

Rake (slip) angle () Degrees

in Figure 3.1. In elastic fault plane theory, fault planeis where violent relative motion

occurs in an earthquake event. Specifically, fault plane is assumed as an rectangular

area of plate interface on foot block and the top and bottom edges of the fault plane are

parallel to the mean Earth surface. The orientation and position of this fault plane are

22


29/65

prescribed by its center location (0, 0), strike direction and dip angle. The center of the

fault plane is called focusand is the location where rupture starts during an earthquake

event. Its projection on the mean surface of the Earth is called epicenter. Focal depthis

defined as the vertical distance between the focus and the epicenter. Strike direction is

defined as the facing direction when someone stands on the top edge of a fault plane with

foot block on his left-hand side and the fault plane on his right-hand side. Then strike

angle,, is the angle measuring clockwise from the local north to the strike direction.

Dip angle,, is the angle between the mean Earth surface and the fault plane, measured

from the mean Earth surface down to the fault plane. The size of the fault plane is

described by its length and width. The length of a fault plane, L, is defined as the

length of its top edge or bottom edge (the edges parallel to the strike direction) and

the widthof the fault plane, W, refers to the length of one of the other two edges. The

above parameters describe the size, location and orientation of a fault plane. The rupture

occurring on this fault plane is described by slip direction (i.e., rake) and dislocation

(amount of slip motion). Slip Direction (rake)describes to which direction the hanging

block moves relative to the foot block on the fault plane. Strike angle, , is the angle

measured counter-clock-wise from strike direction to slip direction on the fault plane.

Dislocation (slip amount)is the distance of motion of hanging block relative to the foot

block along the strike direction on the fault plane.

With the fault parameters defined above, the seafloor deformation can be calculated

via elastic finite fault plane theory. The detailed procedure of this calculation can be

found in the papers by Mansinha and Smylie (1971) and Okada (1985).

Since the floor displacement is evaluated over a plane, special handling is required

to map it onto the mean Earth Surface (i.e., Ellipsoidal surface). Oblique Stereo-

graphic Projection (Snyder, 1987) is implemented in COMCOT to map the surface of

23


30/65

an Ellipsoid (the Earthquake) onto a plane by taking the epicenter as where the plane

is tangential to the Earth surface. For this projection in COMCOT, the Earth is de-

scribed by WGS84 Datum (i.e., semi-major axis a = 6378137.0mand semi-minor axis

b = 6356752.3142mwith scaling factork0 =0.9996). With this projection method, each

location (, ) on the Earth surface (Ellipsoidal surface), corresponds to a point (x,y) on

the plane tangential to the Earth surface at the epicenter (0, 0), whose displacement

can be computed via either Mansinha and Smylies method or Okadas approach.

3.1.2 Transient Seafloor Motion

Different from the instant deformation in which the duration of rupturing process can

be neglected, for a transient seafloor motion, the duration cannot be omitted and should

be considered in numerical simulations. This type of transient seafloor motion could

be the seafloor rupture caused by an earthquake event or submarine landslides. The

inclusion of temporal variation of water depth (or, more exactly, the motion of seafloor)

in Equations (2.1)-(2.13) allows to simulate such type of transient bottom floor motion

as long as long wave assumption is satisfied. This transident seafloor motion model has

been applied to study the transient rupture process of the 2004 Indian Ocean tsunami

(Wang and Liu, 2006).

To simulate tsunamis generated by transient seafloor motion with COMCOT, varia-

tion of water depth in terms of time should be prepared before a simulation starts. The in-

put data file contains a sequence of snapshots of instantaneous seafloor positions relative

to its original position. This snapshot data is then used to evaluate the term ht

in Equa-

tions (2.1)-(2.13). Figure 3.2 shows how water depth changes during a landslide event.

Three snapshots are presented in the sketch. The variations of seafloor in this event are

24


31/65

described by its position relative to its original location. For example, in snapshot t= t2,

the seafloor variations at x = x1,x2,x3 are given as h(x1, t2) = h(x1, t0) h(x1, t2),

h(x2, t2) = h(x2, t0) h(x2, t2) and h(x3, t2) = h(x3, t0) h(x3, t2), respectively. If

h > 0, it means seafloor is being uplifted; ifh < 0, the seafloor is being subsided.

3.2 Water Surface Disturbances

In addition to generating tsunamis via seafloor motions, in COMCOT, tsunamis can also

be generated from a customized initial water surface displacement. When using this

mechanism to generate a tsunami, a data file needs to be prepared which contains the

the profile of water surface displacement measured from the mean surface level. This

displacement will become the source of a tsunami.

25


32/65

h

U

PWARD

0

NORTHIN

G

EASTING

LEGEND

FAULT PLANE (A RECTANGULAR SURFACE ON FOOT BLOCK)

PROJECTION OF FAULT PLANE ON MEAN EARTH SURFACE

DIP ANGLE OF FAULT PLANE

EPICENTER (PROJECTION OF FOCUS ON EARTH SURFACE)

FOCUS OF AN EARTHQUAKE (CENTER OF FAULT PLANE)

SLIP DIRECTION ON FAULT PLANE (RELATIVE TO FOOT BLOCK)

SLIP DIRECTION ON FAULT PLANE (RAKE ANGLE)

STRIKE ANGLE

h FOCAL DEPTH

NORTHIN

G

STRIK

EDIREC

TION

MEAN EARTH SURFACE

MEAN EARTH SURFACE

LW

W

L LENGTH OF FAULT PLANE

WIDTH OF FAULT PLANE

FOOT BLOCK

Figure 3.1: Sketch of a Fault Plane and Fault parameter definitions.

26


33/65

MEAN SEA LEVEL

WATER

SEA BED

WATER

SEA BED

WATER

SEA BED

t = t0

t = t1

t = t2

h(x2,t1)

LEGEND

WATER

SEA BED

SLIDING MATERIALS

ORIGINAL SEA FLOOR (BEFORE LANDSLIDE)

h(x2,t0)

h(x2,t2)

h(x1,t0)

X0 x1 x2

h(x3,t0)

h(x1,t1)

h(x1,t2)

x3

h(x3,t1)

h(x3,t2)

X0 x1 x2 x3

X0 x1 x2 x3

Figure 3.2: Sketch of Transient Sea Floor Motion).

27


34/65

CHAPTER 4

PARAMETER CONFIGURATION, INPUT AND OUTPUT DATA

The user should prepare the topographic information and focal parameters for the spe-

cific tsunami simulation. The detailed information on preparing these files, as well as a

further explanation of the items in input files is given in the follows sections.

4.1 Control Files

COMCOT.CTL defines all the parameters required for tsunami simulations with

COMCOT. If multiple fault planes are implemented, an additional control file,

FAULT MULTI.CTL, is also required to configure parameters for additional fault

planes. These two parameter control files can be edited or viewed with any TXT-editing

applications, such as WordPad, UltraEdit and etc. It should be noted the structure of

these two files CANNOT be altered or changed except for the VALUE FIELDportion.

Otherwise, COMCOT cannot read these two files properly.

In general,COMCOT.CTLcontains 5 parameter sections, which provides control on

the parameters for the simulation, fault model, wave maker, landslides and grid setup.

Each section will be explained in detail in the following sections.

4.1.1 General Information

The first section (see Figure 4.1) in COMCOT.CTL provides the control on the gen-

eral aspects of a numerical simulation, such as the total runtime, initial and boundary

conditions. For each line, parameter value should be specified after :.

28


35/65

Figure 4.1: General Parameter Section in COMCOT.CT L

Total run time defines the total physical duration to be simulated, which should be

given in seconds.

Time interval to Write Data defines the time interval (in seconds) to output data

(including water surface elevation, and volume flux if specified).

Output Zmax &TSis used to control the output of maximum water surface elevation

and time history records at given locations. 0 The maximum water surface elevation

and depression will be output every hour; 1 The the time history records will be output

at up to 9999 locations given in the data file ts location.dat; 2 Both the maximum

water surface elevation and the time history records will be output.

Start Typeis used to specify the starting type of a simulation. 0 Start a simulation

from the very beginning,t= 0 s; 1 Start a simulation from a resuming time step where

data are previously saved.

Resuming Time If hot start specified the moment when a simulation starts from if

29


36/65

hot start is used or, the moment when a snapshot will be saved for a later restart if cold

start is selected.

Specify Min WaterDepth offshoreis used to define the where is the shoreline. Forexample, if a value 10.0 is assigned here, the water depth shallower than 10 .0 meters

will be treated as land region and a vertical wall boundary is set along the 10.0 meter

water depth contour. To specify the true shoreline, 0.0 should be given here which will

be used for runup and inundation calculation.

Initial Cond. (0:FLT,1:File,2:WM,3:LS,4:FLT+LS) is used to define the initial

conditions. 0 use fault model to calculate the seafloor deformation and fault pa-

rameters should be given at fault parameter section in COMCOT.CTL; If multiple

fault planes are implemented, parameters for additional fault planes should be given

in FAULT MULTI.CTL; 1 use customized the profile in an external data file as initial

condition and the file name and format are specified at Fault Parameter section; 2 use

incident wave maker to generate waves; 3 use landslide model to generate tsunamis

and landslide parameters are defined at Landslide section in COMCOT.CTL; 4 use

both fault model and landslide model to generate tsunamis.

Specify BC (0-Open;1-Sponge;2-Wall;3-FACTS) is used to define boundary con-

ditions of the numerical domain. 0 open boundary (i.e., radiation boundary); 1

use sponge layer (absorbing boundary); 2 use vertical wall boundary; 3 use

FACTS data as input boundary condition for far-field tsunami sources and use COM-

COT to calculate nearshore properties. The input boundary conditions are obtained

from the data files given at Specify Input Z filename, Specify Input U filename and

Specify Input V filename.

30


37/65

4.1.2 Parameters for Fault Model

If in the line Initial Cond. (0:FLT,1:File,2:WM,3:LS,4:FLT+LS)in General Param-

eter Section, 0 or 4 is selected, fault parameters should be given in the section

Parameters for Fault Model in COMCOT.CTL. These fault parameters will be used

to calculate seafloor deformation during an earthquake event. Then, it is assumed that

the water surface just mimics the seafloor deformation.

Figure 4.2: Fault Parameter Section inCOMCOT.CT L

Fault parameters for a specific earthquake:

No. of FLT Planes (Use fault multi.ctl if 1) defines how many fault planes will

be implemented to create seafloor deformation. If the number is larger than 1, an ad-

ditional control file, FAULT MULTI.CT Lwill be required to specify parameters for

additional fault planes. The definitions of parameters in FAULT MULTI.CT Lare the

same as this section.

Fault Rupture Time (seconds) defines when this fault plane will rupture. 0.0

means the fault is rupturing at the beginning of the simulation.

Faulting Option (0: Model; 1- Data;) specifies if the seafloor deformation is cre-

31


38/65

ated via fault model (e.g., Okada, 1985) or imported from an external data file. 0 use

fault model with the fault parameters given in this section to calculate seafloor deforma-

tion; 1 use an external data file which contains the seafloor deformation as the initial

condition. The file name and format are given in the last two lines of this section.

Focal Depth is the distance measured vertically from the top edge of a fault plane

to the mean earth surface.

Length of source area is the length of the rectangular fault plane and

Width of source areais the width of the rectangular fault plane.

Dislocation of fault plate defines the relative dislocation between foot block and

hanging block on the fault plane.

Strike direction defines the strike angle () of the fault plane, measured from the

north to the fault line, with the fault plane on the right-hand side.

Dip angleis the angle () between the mean earth surface and the fault plane.

Slip angle is the angle () measured counter clock wise from the fault line to the

direction of relative motion of hanging block on the fault plane.

Comp. Domain- Left Edge - Latitude is the latitude of the lower-left (i.e., south-

east) corner grid in the 1st-level grid,layer01.

Comp. Domain- South Edge - Longitude is the longitude of the lower-left (i.e.,

southeast) corner grid in the 1st-level grid, layer01.

Epicenter Location: Latitude and Epicenter Location: Longitude specify the

latitude and longitude of the epicenter of an earthquake.

32


39/65

File Name of Deformation Dataspecifies the file name of the seafloor deformation

data if 1 is chosen at the line Faulting Option (0: Model; 1- Data;). And the data for-

mat should be given at the line Data Format Option (0-COMCOT; 1-MOST; 2-XYZ).

3 ASCII formats are supported by COMCOT. 0 use the old COMCOT format (ver-

sion1.6 or earlier); 1 use the MOST format and 2 use the XYZ format. Latitude of

the epicenter of an earthquake.

By default, COMCOT will output the initial condition (initial water surface eleva-

tion) in a file named ini sur f ace.dat.

4.1.3 Parameters for Wave Maker

If in the line Initial Cond. (0:FLT,1:File,2:WM,3:LS,4:FLT+LS)in General Param-

eter Section, 2 is selected, parameters for wave maker should be given in the section

Parameters for Wave Makerin COMCOT.CTL. Either solitary waves or a customized

time history profile can be sent into the numerical domain (1st-level grid, layer01)

through any of the four boundaries.

Figure 4.3: Wave Maker Parameter Section in COMCOT.CT L

Wave type ( 1: Solitary, 2:given profile ) is used to determine the incident wave

type: 1 Send Solitary wave into the numerical domain, layer01; 2 Input a wave

profile defined in a time history file through one boundary whose should be given after

33


40/65

FileName of Customized Input (for Type=2); 3 generate a focusing solitary wave.

When using this option, incident wave through a boundary will converge to a specified

location (focus). Two additional parameters will be asked for when the program starts:

x and y coordinates of the focus in layer01.

Incident direction( 1:top,2:bt,3:lf,4:rt,5:ob ) defines which boundary the wave is

sent through. 1 waves come from the top boundary of the domain; 2 waves come

from the bottom boundary of the domain; 3 waves come from the left boundary of the

domain; 4 waves come from the right boundary of the domain; 5 Oblique incident

wave. When using this option, an oblique wave will be sent into the numerical domain

through boundaries. The oblique angle will be required after the program starts. The

angle ranges is measured from the northward (upward) to the incident direction, ranging

from 0.0 to 360..

Characteristic Wave Amplitudespecifies the wave amplitude of the incident wave

(only effective when solitary wave is sent in).

Typical Water depthspecifies the characteristic water depth, which is effective for

both wave types. This value is used to calculate volume flux associated with incident

waves based on shallow water wave theory.

4.1.4 Parameters for Landslides/Transient Floor Motion

If in the line Initial Cond. (0:FLT,1:File,2:WM,3:LS,4:FLT+LS) in General Pa-

rameter Section, 3 or 4 is selected, parameters for submarine landslide model

(or other submarine transient seafloor motions) should be given in the section

Parameters for Transient Sea Floor Motionin COMCOT.CTL. Compared to the di-

34


41/65

mension of the simulated region (i.e., layer01), transient seafloor motions (e.g., transient

fault rupture, landslides), generally, occur within a small confined area, which is defined

by the coordinates of its left (west), right (east), bottom (south) and top (north) margins.

X Coord. of Left/West Edge of Landlide Area specifies X coordinate of grids on

the left/west margin of the landslide region.

x Coord. of Right/East Edge of Landlide Area specifies X coordinate of grids on

the right/east margin of the landslide region.

Y Coord. of Bottom/South Edge of Landlide Areaspecifies Y coordinate of grids

on the bottom/south margin of the landslide region.

Y Coord. of Top/North Edge of Landlide Areaspecifies Y coordinate of grids on

the top/north margin of the landslide region. It should be noted that the units/coordinates

used here should be the same as those of 1st-level grid. For example, if the 1st-level grid,

layer01, adopts Spherical Coordinates (Latitude/Longitude), then the values defining the

landslide region should also be latitude/longitude in Spherical Coordinates.

File Name of SeaFloor Motion Dataspecifies the filename of seafloor motion data.

This data file contains a time sequence of snapshots of variation of the seafloor relative

to its original location.

4.1.5 Parameters for All Grid Levels

Parameters for1st-Level Grid

In COMCOT.CTL, parameter section for grids of all levels follows the parameter sec-

tion of landslide model. The section, Parameters for 1st-level grid layer 01, con-

35


42/65

tains configurations for the 1st-level grid (the largest grid region), layer01.

Figure 4.4: Parameters for 1st-level grid inCOMCOT.CT L

The meaning of each entry is illustrated as follows.

Run This Layer?is used to determine if this grid level is included in the numerical

simulation. 0 this grid level will be include; 1 this grid level will not be included in

the simulation.

Coordinate Systemspecifies which coordinates will be used for this grid level. 0

use Spherical Coordinates; 1 use Cartesian Coordinates.

Governing Equations determines which type of governing equations will be used

for this grid level. 0 use linear shallow water equations; 1 use nonlinear shallow

water equations. If runup and inundation are going to be calculated, nonlinear shallow

water equations must be chosen.

Grid Size specifies the grid size (x) adopted for this grid level. If spherical co-

ordinate system is adopted, the grid size should be given in arc minutes; if cartesian

coordinate system is adopted, the grid size should be given in meters.

Time Step determines the time step (t) used for the numerical simulation. The

36


43/65

time step chosen should guarantee that Courant condition is satisfied, which means

t< x

ghmax

(4.1)

where g is the gravitational acceleration and hmax is the maximum water depth within

the region of this grid level. If Courant condition is not satisfied for a given time step

(t), COMCOT will automatically adjust it to make Courant condition being satisfied.

Inside the code, the maximum time step size tallowed is fixed as 0.5x/

ghmax.

Bottom Friction Switch? is used to determine if bottom friction is considered in

the numerical simulation. 0 bottom friction will be included and a constant bot-

tom friction coefficient will be applied (Mannings n) and the roughness is given at

Mannings Roughness Coef.; 1 bottom friction will not be considered; 2 bot-

tom friction will be included and variable Mannings roughness coefficient will be

applied. The variable roughness coefficients need to be given in a data file named

f ric coe f layerX X.datin which X Xrepresents the grid ID of this grid level.

Ouput Option? (0: Z+Hu+Hv; 1: Z Only; 2: NONE) is used to control the out-

put of free surface elevation and volume fluxes. 0 both water surface elevation and

volume fluxes (i.e., product of water depth and velocity) will be output; 1 only water

surface elevation not be output. 2 both water surface elevation and volume fluxes will

not be output.

In COMCOT, a grid region (i.e., a rectangular area) is defined by the coordinates

of its lower-left /south-west corner (X Start,Y Start) and upper-right/north-east corner

(X End,Y End).

X Startspecifies X coordinate of the left/west margin of the grid region.

X Endspecifies X coordinate of the right/east margin of the grid region.

37


44/65

Y Startspecifies Y coordinate of the bottom/south margin of the grid region.

Y Endspecifies Y coordinate of the top/north margin of the grid region.

Together with the given grid size, the water depth/land elevation at every grid will

be extracted from the input bathymetry/topography data file whose name is given af-

terFile Name of Bathymetry Data. File Name of Bathymetry Dataspecifies the file-

name of bathymetry data for this grid level. This data file contains the gridded bathy-

metric/topographic data for this grid level.

Data Format Optionis used to determine which type of data format is adopted for

bathymetry data file. 0 use old COMCOT data format (version 1.6 or earlier); 1

use MOST format; 2 use XYZ format (3-column data containing x, y and bathymetry

information); 3 use ETOPO bathymetry data. XY Zformat is suggested.

Grid Identification Number is used to assign an ID to this grid level. This is the

only identification number used to distinguish this grid from the others. It is not sug-

gested to change the value here.

Grid Levelis used to determine the level of this grid in the hierarchy of nested grid

setup. For the largest grid layer (the 1st level), the grid level is 1.

Parent Grids ID Number is used to specify which grid layer is the parent grid of

this grid layer. The parent grid means the grid layer where the current grid is directly

nested in. For the 1st-level grid, the parent grid ID should be specified as -1 (or 0) since

it has no parent grid.

38


45/65

Parameters for sub-level grids

The parameter configurations for sub-level grids are very similar to that of the 1st-level

grid. But, there is no need to specify the grid size (x) and time step (t) for sub-level

grids. However, the grid size ratio of its parent grid to this grid level should be given

after GridSize Ratio of Parent layer to current layer. The grid size ratio can be any

integer larger than or equal to 1. However, a ration less than 10 is suggested.

Figure 4.5: Parameters for sub-level grids inCOMCOT.CT L

In COMCOT, a sub-level grid region (i.e., a rectangular area) is defined by the coor-

dinates of its lower-left/south-west corner (X Start,Y Start) and upper-right/north-east

corner (X End,Y End). This four values should be given in the same coordinates as

that used in its parent grid layer. If the current grid layer adopts Cartesian coordinates

(UTM) and its parent grid layer adopts Spherical Coordinates (latitude/longitude), then

the coordinates of these two corners should be given in laitutde/longitude. In COMCOT,

a sub-level grid layer using Cartesian coordinates can be nested inside a grid layer using

Spherical coordinates. However, the reverse nesting is not supported.

39


46/65

4.2 Input Data

4.2.1 Bathymetric/Topographical Data

4 ASCII formats of bathymetry data are supported in COMCOT v1.7, the old COMCOT-

formatted data (for version 1.6 or earlier), MOST-formatted data and XYZ format data

and ETOPO bathymetry data. However, XYZ format is suggested whose format will

be detailed here. The bathymetry data file should be in ASCII format and written in 3

columns. The first column contains information of X coordinates (rightward, pointing to

the east), and second column contains information of Y coordinates (upward, pointing to

the North) and the third column contains the data of water depth (positive for bathymetry

data and negative for topographical data), see Figure 4.6.

Figure 4.6: Example of Bathymetry Data File (XYZ format)

The format of ETOPO data is the same as that of XYZ 3-column data. The only

difference is the sign of data. In ETOPO data, the value of water depth is given as

a negative number and the land elevation is given as a positive number. The ETOPO

40


47/65

bathymetry data (either ETOPO1 or ETOPO2) can be directly used in COMCOT v1.7.

COMCOT will change the sign after the ETOPO data is loaded in.

The bathymetry data must be gridded and the gridded data (a nx nymatrix) should

be written row by row or column by column from the left/west (i = 1) to the right/east

(i=n x) from the bottom/south (j = 1) to the top/north (j=n x) wherei and jrepresent

the x, y grid index of bathymetry/topography grids and n xand ny are the total number

of grids in X and Y direction. The x coordinates must increase monotonically and y

coordinates should increase or decrease monotonically. It should be noted that for each

grid layer the gridded bathymetry/topography data should adopt the same coordinate

system as specified for this grid layer. And the area of bathymetry/topography data

should be larger than or equal to the region defined by X S tart, X End, Y Start and

Y Endin the parameter section for this grid layer.

If a grid layer adopts Cartesian coordinate system, however, its parent grid layer

uses Spherical Coordinate system, X Start, X End, Y Start and Y End of this grid

layer should given with spherical coordinates (latitude, longitude). But, the bathymetry

data file of this grid layer should be prepared with cartesian coordinates (UTM). The

position of this grid layer is determined by X Start, X End,Y S tartand Y Endwhich

are given in longitude and latitude. The key to remember is the position of a sub-level

grid layer is also defined in the same coordinates as its parent grid layer.

4.2.2 Seafloor Deformation Data

Similar to bathymetry/topography data file, 3 ASCII formats of seafloor deformation

data are also supported in COMCOT v1.7, the old COMCOT-formatted data (for version

1.6 or earlier), MOST-formatted data and XYZ format data. However, XYZ format is

41


48/65

also suggested. The seafloor deformation data file should be in ASCII format and written

in 3 columns. The format is the same as that of XYZ-format bathymetry data file. The

first column contains information of X coordinates (rightward, pointing to the east), and

second column contains information of Y coordinates (upward, pointing to the North)

and the third column contains the data of water depth (positive for bathymetry data and

negative for topographical data), see Figure 4.7.

Figure 4.7: Example of Seafloor Deformation Data File (XYZ Format)

The deformation data should be gridded and written row by row from the left/west

(i = 1) to the right/east (i = n x) from the bottom/south (j = 1) to the top/north (j =n x)

whereiand jrepresent thex,ygrid index of bathymetry/topography grids andnxandny

are the total number of grids inXand Ydirection. It should be noted that the deformation

data should use the same coordinate system as the 1st-level grid. The area of deformation

data should fall inside the region defined by the 1st-level grid.

42


49/65

4.2.3 Transient Sea Floor Motion Data

To generate tsunamis by landslides or any other type of transient seafloor motion, a

data file is required to provide information on seafloor motion. This data file contains a

sequence of snapshots of seafloor variations in terms of time.

If, for example, the landslide area has a grid dimension ofn xand ny where n xand

ny stand for the total number of grids in X and Ydirection, respectively, and in total,

ntsnapshots are created to trace the variation of seafloor position in time, the variation

of seafloor can be represented as h(i, j, k), where i = 1, nx, j = 1, ny and k = 1, nt.

h(i, j, k) describes the variation of the seafloor position against its origin at (i, j) at time

levelk. And h(i, j, k) can be obtained from

h(i, j, k) = h(i, j, k) h0(i, j) (4.2)

in which h(i, j, k) denotes the instantaneous water depth at (i, j) at time level k and

h0(i, j) represents the original position of seafloor.

Figure 3.2 gives a simply example of seafloor variation during a landslide event.

3 snapshots are presented to show the positions of seafloor during a landslide event.

Snapshot att=t0 shows the original position of sea floor before landslide and snapshots

at t = t1 and t = 2 indicate the intermediate and final stage of seafloor position. Then

the variation of seafloor can be described by these 3 snapshots. For example, at x =

x1, 3 snapshots of seafloor variation are h(x1, t0), h(x1, t1) and h(x1, t2) where

h(x1, t0) = h(x1, t0) h(x1, t0), h(x1, t1) = h(x1, t0) h(x1, t1) and h(x1, t2) =

h(x1, t0)h(x1, t2), respectively. Ifh > 0, it indicates that the seafloor is being uplifted;

ifh < 0, it means the seafloor is being subsided.

After the snapshot data of seafloor variation is determined, the input data file for

this transient seafloor motion can be written in the following format. The first line of

43


50/65

the data file contains 3 integers specifying the grid dimension, n xand ny, and the total

number of snapshots, nt. Then, it follows by the Xcoordinates of gridded data (from

lines 2 tonx + 1), Ycoordinates of gridded data (from lines nx + 2 tonx + ny + 1) and

the times at which the snapshots are created (from linesnx + ny + 2 to nx + ny + nt+ 1).

After the coordinates and time information, it is the section for snapshot data. All thent

snapshots of seafloor variation are written into a single column one by one from t = t1

tot = tnt. For each snapshot, e.g., h(nx, ny, t1), the data should be written row by row

from the left/west (i = 1) to the right/east (i = nx) from the bottom/south (j = 1) to

the top/north (j = nx) where i and j represent the x, y grid index of gridded snapshot

data. It should be noted that the landslide snapshot data should use the same coordinate

system as the 1st-level grid. The landslide area should fall inside the region defined by

the 1st-level grid. However, the grid and time resolutions are not necessarily the same

as those of the 1st-level grid layer. Interpolation will be carried out inside COMCOT. A

sample data file is shown in Figure??.

Figure 4.8: Example of Landslide Data File (XYT Format)

Two file formats are supported: the old COMCOT format (for version 1.6) and the

new XYT format. The new format is suggested for version 1.7.

44


51/65

4.2.4 Bottom Friction Coefficients

If at the grid parameter section, 2 is chosen afterBottom Friction Switch?for one grid

layer, a gridded Mannings roughness n (varying in space) should be prepared for this

grid layer before simulation starts. The roughness coefficient data file has the same

format as that of bathymetric/topographical data. The grid size is not necessarily the

same as that of the grid layer, but the area covered should be slightly larger than or

at least equal to the grid layer. The file name of roughness coefficients should be like

f ric coe f layerX X.datin which X Xrepresents the grid ID of this grid layer.

4.2.5 Time History Input for Wave Maker

A time history input of water surface elevation is required if the wave maker is im-

plemented and the option 2 is chosen for Wave type ( 1: Solitary, 2:given profile ).

The data file should be written in ASCII format and the file name need to be given

at the line FileName of Customized Input (for Type=2). The data file contains two

data columns. The first column contains the time sequence, t = tk (k = 1, , nt),

and the second column contains the water surface elevation, = k (k=1, nt), corre-

sponding to the time sequence. The units should besecondsfor time andmetersfor the

water surface elevation.

4.2.6 Numerical Tidal Gauge Locations

If in the control file COMCOT.CTL, Output Zmax & TS is set to 1 or 2, the the time

history records will be output at up to 9999 numerical tidal gauge location. A data file,

textb f ts location.dat, is required to specify these tidal gauge locations. The data file

45


52/65

contains two data columns. The first data column defines the X coordinates of tidal

gauges and the second column defines the Ycoordinates of tidal gauges. In Figure 4.9,

locations of four tidal gauges are specified.

Figure 4.9: Example of Tidal Gauge Location File ts location.dat

4.3 Output Data

4.3.1 Time Sequence Data

A data file, named time.dat, will always be output for a simulation. The data file con-

tains only one data column, which is the wall-clock time corresponding to each time

step during the entire simulation. This output file can be used together with the time

history output at specified tidal gauges to create time history plots.

4.3.2 Water Surface Elevation/Volume Fluxes

Generally, only the free surface elevation will be output in data files named in the form

z ID xx xx xx.dat, wherezstands for free surface elevation, two digits I Ddenote the cor-

responding grid identification number and the six digits,xxxxxx, corresponds to the time

step number when the data is output. Therefore, the data file z ID xxxxxx.datstores the

46


53/65

free surface elevation at all grid points in the grid layer. For example, z 01 001234.dat

stores the free surface elevation at all the grid points of layer01 at time step 1234. It

should be mentioned that the total number of time steps required for a simulation and

also time step number for data output are also calculated based on the time step (t)

of layer01. For example, if 0.5s time step is used for layer01, the number of time

steps, 1234, corresponds to a physical time t = 1234 0.5 = 617.0s. However, if

the switch Output Volume Flux is set to 0 for this layer, two additional data files,

m ID xxxxxx.datand n ID xxxxxx.datwill be created, storing volume flux data in x

direction andydirection, respectively, at all the grid points in this grid layer at time step

number xxxxxx. The following scripts will be able to read the data in a output data file

F AB dddddd.dat into a predefined matrix B(nx,ny)

Table 4.1: Fortran scripts to write Water surface/Volume flux data into a data file

open (IO-UNIT,file=z ID xxxxxx.dat,status=unknown)

do i=1, ny

read (IO-UNIT, (15f9.3)) (B(i, j), i=1, nx)

enddo

close(IO-UNIT)

In Table4.1,zdenotes water surface elevation, for volume flux,z should be replaced

by m for volume flux in X direction orn for volume flux in Y direction; ID represents

Identification Number of a grid layer and xxxxxx stands for the number of time steps

at which the data is written. In addition, if the switch Output Volume Fluxis set to 2,

neither water surface elevation nor volume fluxes will be out.

To obtain the velocity components (u, v) at a grid, divide the volume flux components

by the total water depth ( + h) where represents the water surface fluctuation.

47


54/65

4.3.3 Initial Condition

At the beginning of a simulation starts, the initial water surface displacement will be

written into a data file named ini sur f ace.dat. This data file is written via the following

FORTRAN code.

Table 4.2: Fortran scripts to write Initial Condition into a data file

open (IO-UNIT,file=ini surface.dat,status=unknown)

do i=1, ny

write (IO-UNIT, (15f8.3)) (deform(i, j), i=1, nx)

enddo

close(IO-UNIT)

In Table 4.2, deform(i, j) denotes initial water surface fluctuation at grid (i, j); nx

andny denotes the total number of grids in Xand Ydirections, respectively.

4.3.4 Seafloor Deformation

If multiple fault planes are implemented,i.e., in COMCOT.CT L,Initial Cond. (0:FLT,1:File,2:W

is set to 0 or 4 and No. of FLT Planes (Use fault multi.ctl if 1) is larger than 1, the

seafloor deformation generated by each fault plane will be output in data files named

deform segXX.datwhere xxdenotes the numbering of fault planes, starting from 01 up

to 99. This type of data file is written via the following FORTRAN code.

In Table 4.3, deform(i, j) denotes seafloor deformation at grid (i, j) generated by

fault plane xx; nx and ny denotes the total number of grids in X and Y directions, re-

spectively.

48


55/65

Table 4.3: Fortran scripts to write Seafloor Deformation into a data file

open (IO-UNIT,file=deform segXX.dat,status=unknown)

do i=1, ny

write (IO-UNIT, (15f8.3)) (deform(i, j), i=1, nx)

enddo

close(IO-UNIT)

4.3.5 Maximum Water Surface Elevation/Depression

If inCOMCOT.CT L, Output Zmax & TS is set to 0 or 2, the maximum water surface

elevation/depression will be output every hour and at the end of a simulation. Every

one hour during the simulation, two data files will be created for each grid layer, named

etamax layerID XXXXhrs.datand etamin layerID XXXXhrs.dat, in which IDde-

notes the two-digit identification number of each grid layer and the 4-digit XXXXrepre-

sents the number of wall-clock hours elapsing since the tsunami event occurs. The data

file etamax layerID XXXXhrs.dat contains the maximum water surface elevation at

every grid for each grid layer included in the simulation in the past XXXXhours and

etamin layerID XXXXhrs.datcontains the maximum water surface depression at ev-

ery grid for each grid layer included in the simulation in the past XXXX hours. For

example, etamax layer01 0001hrs.datstores the maximum water surface elevation at

all the grid points oflayer01 within the past 1.0 hours andetamin layer01 0001hrs.dat

stores the maximum water surface depression at all the grid points oflayer01 within the

past 1.0 hours. The data file, for example, etamax layer01 0001hrs.dat, is written via

the following FORTRAN code.

In Table4.4,max denotes the maximum water surface elevation of grid layer 01; nx

andnyare the total numbers of grids in Xand Ydirections. At the end of the simulation,

49


56/65

Table 4.4: Fortran scripts to write Max/Min Water Surface Fluctuations into a data

file

open (IO-UNIT,file=etamax layer01 0001.dat,status=unknown)

do i=1, ny

read (IO-UNIT, (15f9.4)) (max(i, j), i=1, nx)

enddo

close(IO-UNIT)

two data files,etamax layer01.datandetamin layer01.dat, will also be created storing

the maximum water surface elevation/depression within the entire simulated duration. It

should be mentioned that this type of maximum water surface elevation/depression data

files will be created for every grid layer included in the numerical simulation.

4.3.6 Time History Records

If in the control file COMCOT.CTL, Output Zmax & TS is set to 1 or 2, the the

time history records will be output at up to 9999 locations given in the data file

ts location.dat. A series of data files will be created with the names in the for-

mat like ts recordxxxx.dat in which xxxx represents the position of a tidal gauge

in the data file ts location.dat. For example, if in total 4 tidal gauges are de-

fined in ts location.dat, then four data files, ts record001.dat, ts record0002.dat,

ts record0003.dat and ts record0004.dat, will be created corresponding to the four

tidal gauges, numbering from top to bottom, in ts location.dat.

50


57/65

4.3.7 Hot Start/Automatic Data Backup

By default, COMCOT will automatically create a resuming snapshot during a sim-

ulation for later hot start. The resuming snapshot will be created at the mo-

ment specified at Resuming Time If hot start in COMCOT.CTL. The simulation

can restart from this save snapshot. For this snapshot, three data files will

be created which are z1 layerID XXXXXX.dat, m1 layerID XXXXXX.dat and

n1 layerID XXXXXX.dat. In the file names, ID denotes the 2-digit identification

number of a grid layer and XXXXXX denotes the corresponding 6-digit time step

number. For example, ifResuming Time If hot start is set to 1000.0 in a simu-

lation including only one grid layer, 01 and the time step t = 1.5, at time step

number NINT(1000.0/t) = 667 (NINT means the nearest integer), the following

three data files will be created: z1 layer01 000667.dat, m1 layer01 000667.dat and

n1 layerlayer01 000667.dat. The data in these data files are all written in the follow-

ing format (FORTRAN script) in Table4.5.

Table 4.5: Fortran scripts to write snapshots for hot start

open (

Documents

COMCOT User Manual v1.7